This page contains all questions and answers asked by students from this class. The most recent questions will be at the top.
![Rendered by QuickLaTeX.com LHS = \sum_{r=2}^n \frac{1}{(r+1)(r-1)}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-47bc88fcd55867c687a3f66f6cc4319b_l3.png)
![Rendered by QuickLaTeX.com = \sum_{r=2}^n \frac{1}{2} (\frac{1}{r-1} - \frac{1}{r+1})](http://theculture.sg/wp-content/ql-cache/quicklatex.com-114054ca3f6ec5e3e36ca8062f0104f4_l3.png)
![Rendered by QuickLaTeX.com = \frac{1}{2} \sum_{r=2}^n \frac{1}{r-1} - \frac{1}{r+1}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-75a67a9989312e2d0d88d3b4e1a3b102_l3.png)
![Rendered by QuickLaTeX.com = \frac{1}{2} [\frac{1}{1} - \frac{1}{3}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-23139fe518658a1ea819632d0b0aa218_l3.png)
![Rendered by QuickLaTeX.com + \frac{1}{2} - \frac{1}{4}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-22c1dd35e019879e09dbfacc123f0da7_l3.png)
![Rendered by QuickLaTeX.com + \frac{1}{3} - \frac{1}{5}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-effca8277de89404f420596c4ef0e421_l3.png)
…
![Rendered by QuickLaTeX.com + \frac{1}{n-3} - \frac{1}{n-1}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-938e8dc3f7c0aa1b4bfa39ad6815f932_l3.png)
![Rendered by QuickLaTeX.com + \frac{1}{n-2} - \frac{1}{n}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-fe8c57371838a44b9ca81a20295c50ff_l3.png)
![Rendered by QuickLaTeX.com + \frac{1}{n-1} - \frac{1}{n+1}]](http://theculture.sg/wp-content/ql-cache/quicklatex.com-8a37d2ae700ebdd6931c5eb7d00f8a5a_l3.png)
![Rendered by QuickLaTeX.com = \frac{1}{2} [\frac{3}{2} - \frac{1}{n} - \frac{1}{n+1}]](http://theculture.sg/wp-content/ql-cache/quicklatex.com-9b2f7b59e500a86896494ad950404aeb_l3.png)
![Rendered by QuickLaTeX.com = \frac{3}{4} - \frac{1}{2n} - \frac{1}{2n+2}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-15916f930d5f52f9dd8d28376cc44848_l3.png)
Since the
is a constant, the series convergences.
The sum to infinity
Let
![Rendered by QuickLaTeX.com T_n](http://theculture.sg/wp-content/ql-cache/quicklatex.com-8989e708af6cb8a6919e0c9890a57342_l3.png)
![Rendered by QuickLaTeX.com n^{th}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-2abf0b71b2242df3a72e948f3e5c9ea0_l3.png)
![Rendered by QuickLaTeX.com T_n = a + (n-1)d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-76ed1a6910ea648f99700094e863fa6a_l3.png)
![Rendered by QuickLaTeX.com T_1 = a](http://theculture.sg/wp-content/ql-cache/quicklatex.com-4a4ab15e118e9178fea974e64c2e4186_l3.png)
![Rendered by QuickLaTeX.com T_3 = a +2d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f64815649448880419efb0d2920e4138_l3.png)
![Rendered by QuickLaTeX.com T_6 = a +5d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-235fddec468e6d142858926af2ca81ec_l3.png)
Since they are consecutive terms of a GP,
![Rendered by QuickLaTeX.com \frac{T_3}{T_1} = \frac{T_6}{T_3} = r](http://theculture.sg/wp-content/ql-cache/quicklatex.com-84aabe8256f55cc77e77f68c4562ae60_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow \frac{a+2d}{a} = \frac{a+5d}{a+2d}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-554ffc0949ef7a2c65cf96238ada2b70_l3.png)
![Rendered by QuickLaTeX.com (a+2d)^2 = a(a+5d)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f5dc7b83511339e6ab1cde992503a8d6_l3.png)
![Rendered by QuickLaTeX.com a^2 + 4ad + 4d^2 = a^2 + 5ad](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d5f9501ed0991bc5a2aaab2d995a7244_l3.png)
![Rendered by QuickLaTeX.com 4d^2 - ad =0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-ab4a3bf4cd878fde7e5a6d05599721b8_l3.png)
![Rendered by QuickLaTeX.com d(4d - a) = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-cf9800511192351faebd44cc522cbd84_l3.png)
![Rendered by QuickLaTeX.com d = 0 (NA) \mathrm{~or~} 4d = a](http://theculture.sg/wp-content/ql-cache/quicklatex.com-03b42140205b0fe219a3a3781078978f_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow r = \frac{a+2d}{a}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-81451065ad4d09438bcf62c70a6348c6_l3.png)
![Rendered by QuickLaTeX.com r = \frac{6d+2d}{4d} = \frac{3}{2} > 1](http://theculture.sg/wp-content/ql-cache/quicklatex.com-c5dc7c232b4c1373405fa95198866d21_l3.png)
Sum of first 3 terms
![Rendered by QuickLaTeX.com = \frac{3}{2} (2a+(3-1)d)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-6a63e917199d8849bf46ad116ada9dbf_l3.png)
![Rendered by QuickLaTeX.com 6 = 3a+3d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-a4efb9de258452a9870015c6068d7a86_l3.png)
![Rendered by QuickLaTeX.com a+d=2](http://theculture.sg/wp-content/ql-cache/quicklatex.com-2bb4073be55fc4cb2b2fba82242913a9_l3.png)
Sum of last 3 terms
![Rendered by QuickLaTeX.com = \frac{3}{2}[2(a+(n-1)d) + (3-1)(-d)]](http://theculture.sg/wp-content/ql-cache/quicklatex.com-5907ba072b3a623909edf4ccb307d79a_l3.png)
![Rendered by QuickLaTeX.com T_n = a + (n-1)d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-76ed1a6910ea648f99700094e863fa6a_l3.png)
![Rendered by QuickLaTeX.com -d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-81a52a0a2b2c4c1de3c4e14b3623cd69_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow 231 = \frac{3}{2}(2a + 2nd - 2d -2d)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-c078e98d43a2f9134ae5f776f527e8c6_l3.png)
![Rendered by QuickLaTeX.com 231 = 3a + 3nd - 6d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-62806c264b44fbed37c18a77486b70e7_l3.png)
![Rendered by QuickLaTeX.com 77 = a + nd -2d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-76200e21aeb5d5e687438b2e7f263f4d_l3.png)
Sum of n terms =
![Rendered by QuickLaTeX.com \frac{n}{2}[2a+ (n-1)d]](http://theculture.sg/wp-content/ql-cache/quicklatex.com-903b915470594a8ff6e4fe238f69834f_l3.png)
![Rendered by QuickLaTeX.com 1106 = \frac{n}{2}(2a + nd - d)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-149c040b99524f18a8a2425411cf2198_l3.png)
Solve for n.
(i) Volume,
![Rendered by QuickLaTeX.com V = \pi r^2 h](http://theculture.sg/wp-content/ql-cache/quicklatex.com-b9a366c595eb865d8781c1e91ff4747f_l3.png)
Volume of kth later,
![Rendered by QuickLaTeX.com V_k = \pi [(20)(\frac{5}{6})^{k-1}]^2(22)(\frac{4}{5})^{k-1}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-6d8a0f98c6bda03eab99b46802bed1ac_l3.png)
![Rendered by QuickLaTeX.com V_k = 8800 \pi [\frac{25}{36} \times \frac{4}{5}]^{k-1}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-0c19604d708928bbb2a7636a76ca997e_l3.png)
![Rendered by QuickLaTeX.com V_k = 8800 \pi (\frac{5}{9})^{k-1}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d0d3b9f5377c95c9e40e9853cc4a7d09_l3.png)
(ii)
Since
![Rendered by QuickLaTeX.com r = \frac{5}{9} <1, S_{\infty}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-2ab3b90aa06bb7d756fdc8f6047b43f9_l3.png)
Theoretical Max Volume,
![Rendered by QuickLaTeX.com S_{\infty} = \frac{8800 \pi}{1 - \frac{5}{9}} = 19800 \pi](http://theculture.sg/wp-content/ql-cache/quicklatex.com-51f90b0cf70a7aabf0c42d9d716a8672_l3.png)
Total Volume,
![Rendered by QuickLaTeX.com S_n = \frac{8800 \pi (1 - (\frac{5}{9})^n)}{1 - \frac{5}{9}}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-ea9d92d65dffab5e10332a0bbcd788cd_l3.png)
We want
![Rendered by QuickLaTeX.com S_n \le 0.95 S_{\infty}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-eb87adc71e0850ce5e569c4742a2bdb9_l3.png)
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